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The generalized wreath product of permutation groups is introduced. By meansof it we study the schurity problem for S-rings over a cyclic group $G$ and theautomorphism groups of them. Criteria for the schurity and non-schurity of thegeneralized wreath product of two such S-rings are obtained. As a byproduct ofthe developed theory we prove that $G$ is a Schur group whenever the totalnumber $\Omega(n)$ of prime factors of the integer $n=|G|$ is at most 3.Moreover, we describe the structure of a non-schurian S-ring over $G$ when$\Omega(n)=4$. The latter result implies in particular that if $n=p^3q$ where$p$ and $q$ are primes, then $G$ is a Schur group.

Schurity of $S$-rings over a cyclic group and generalized wreath product of permutation groups